Transcript mathpower
Chapter 1 Systems of Equations 1.3 and 1.5 MATHPOWERTM 11, WESTERN EDITION 1.1 Problem 1: The sum of two numbers is 16. Twice the first number plus three times the second number is 34. Find the two numbers. 2x + 2y = 32 Let x = the first number. 2x + 3y = 34 Let y = the second number. -y = -2 x + y = 16 (1) y=2 2x + 3y = 34 (2) 2x + 3y = 34 2x + 3(2) = 34 Check: 2x = 28 x + y = 16 x = 14 14 + 2 = 16 16 = 16 Therefore, the numbers are 14 and 2. 1.2 Problem 2: Mike has 20 coins, in dimes and quarters. If the total value is $2.75, how many coins of each type are there? 10d + 10q = 200 Let d = # of dimes. 10d + 25q = 275 Let q = # of quarters. -15q = -75 q=5 d + q = 20 (1) 10d + 25q = 275 (2) 10d + 25q = 275 Check: 10d + 25(5) = 275 d + q = 20 d = 15 15 + 5 = 20 20 = 20 Therefore, Mike has 15 dimes and 5 quarters. 1.3 Problem 3: Sue invested $2000, part at 10% and part at 12%. If the total interest earned was $216, how much did she invest at each rate? 10x + 10y = 20 000 Let x = amount at 10%. 10x + 12y = 21 600 Let y = amount at 12%. -2y = -1 600 y= 800 x + y = 2000 (1) 10x + 12y = 21600 (2) 10x + 12y = 21 600 Check: 10x + 12(800) = 21 600 x + y = 2000 10x = 12 000 1200 + 800 = 2000 x = 1 200 2000 = 2000 Sue invested $1200 at 10% and $800 at 12%. 1.4 Problem 4: It is a 230 km trip to the Jackson’s cabin. Part of the trip is on gravel roads, where they travel at a rate of 50 km/h, and part is on paved roads, where they travel at 80 km/h. If the total trip is 4 h, how much time is spent on gravel roads? Let x = time on gravel. 50x + 50y = 200 Let y = time on paved. 50x + 80y = 230 x+y=4 (1) 50x + 80y = 230 (2) Check: x+y=4 3+1=4 4=4 -30y = -30 y= 1 50x + 80y = 230 50x + 80(1) = 230 50x = 150 They travelled 3 h x= 3 on gravel roads. 1.5 Problem 5: A butcher has supplies of lean beef with 15% fat and fat trim that is 100% fat. How many kilograms of lean beef and fat trim does she need to make 50 kg of hamburger, which is 25% fat? Let x = # of kg of beef. Let y = # of kg of trim. x + y = 50 (1) 15x + 100y = (25) x 50 (2) Check: x + y = 50 44.11 + 5.89 = 50 50 = 50 15x + 15y = 750 15x + 100y = 1250 -85y = -500 y= 5.89 15x + 100y = 1250 15x + 100(5.89) = 1250 x = 44.11 Therefore, she needs 44.11 kg of beef and 5.89 kg of trim. 1.6 Problem 6: A fishing boat put out to sea in the morning travelling with the tide. It took 20 min to cover the 6 km to the captain’s favourite fishing grounds. The return trip was against the tide and took 36 min. What was the speed of the boat in still water and the speed of the tide? Let x = speed of the boat. Let y = speed of the current. x + y = speed with the current x - y = speed against the current Recall: d = st 20 x y 6 60 36 x y 6 60 (1) (2) 1 x y 6 3 x + y = 18 x + y = 18 x - y = 10 2x = 28 x = 14 3 x y 6 5 3x - 3y = 30 x - y = 10 14 - y = 10 y=4 The speed of the boat in still water: 14 km/h The speed of the tide: 4 km/h 1.7